Optimal. Leaf size=198 \[ \frac{3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac{a^6}{b^4 (a+b x) (b c-a d)^3}-\frac{3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}-\frac{x (2 a d+3 b c)}{b^3 d^4}+\frac{2 c^5 (2 b c-3 a d)}{d^5 (c+d x) (b c-a d)^3}-\frac{c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac{x^2}{2 b^2 d^3} \]
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Rubi [A] time = 0.266823, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ \frac{3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac{a^6}{b^4 (a+b x) (b c-a d)^3}-\frac{3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}-\frac{x (2 a d+3 b c)}{b^3 d^4}+\frac{2 c^5 (2 b c-3 a d)}{d^5 (c+d x) (b c-a d)^3}-\frac{c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac{x^2}{2 b^2 d^3} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{x^6}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac{-3 b c-2 a d}{b^3 d^4}+\frac{x}{b^2 d^3}+\frac{a^6}{b^3 (b c-a d)^3 (a+b x)^2}+\frac{3 a^5 (-2 b c+a d)}{b^3 (b c-a d)^4 (a+b x)}+\frac{c^6}{d^4 (-b c+a d)^2 (c+d x)^3}+\frac{2 c^5 (2 b c-3 a d)}{d^4 (-b c+a d)^3 (c+d x)^2}+\frac{3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{d^4 (-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{(3 b c+2 a d) x}{b^3 d^4}+\frac{x^2}{2 b^2 d^3}-\frac{a^6}{b^4 (b c-a d)^3 (a+b x)}-\frac{c^6}{2 d^5 (b c-a d)^2 (c+d x)^2}+\frac{2 c^5 (2 b c-3 a d)}{d^5 (b c-a d)^3 (c+d x)}-\frac{3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac{3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 0.257415, size = 198, normalized size = 1. \[ \frac{3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac{a^6}{b^4 (a+b x) (b c-a d)^3}+\frac{3 a^5 (a d-2 b c) \log (a+b x)}{b^4 (b c-a d)^4}-\frac{x (2 a d+3 b c)}{b^3 d^4}+\frac{6 a c^5 d-4 b c^6}{d^5 (c+d x) (a d-b c)^3}-\frac{c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac{x^2}{2 b^2 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 264, normalized size = 1.3 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{2}{d}^{3}}}-2\,{\frac{ax}{{d}^{3}{b}^{3}}}-3\,{\frac{cx}{{b}^{2}{d}^{4}}}-{\frac{{c}^{6}}{2\,{d}^{5} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+15\,{\frac{{c}^{4}\ln \left ( dx+c \right ){a}^{2}}{{d}^{3} \left ( ad-bc \right ) ^{4}}}-18\,{\frac{{c}^{5}\ln \left ( dx+c \right ) ab}{{d}^{4} \left ( ad-bc \right ) ^{4}}}+6\,{\frac{{c}^{6}\ln \left ( dx+c \right ){b}^{2}}{{d}^{5} \left ( ad-bc \right ) ^{4}}}+6\,{\frac{{c}^{5}a}{{d}^{4} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-4\,{\frac{{c}^{6}b}{ \left ( ad-bc \right ) ^{3}{d}^{5} \left ( dx+c \right ) }}+{\frac{{a}^{6}}{{b}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}+3\,{\frac{{a}^{6}\ln \left ( bx+a \right ) d}{{b}^{4} \left ( ad-bc \right ) ^{4}}}-6\,{\frac{{a}^{5}\ln \left ( bx+a \right ) c}{{b}^{3} \left ( ad-bc \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28764, size = 738, normalized size = 3.73 \begin{align*} -\frac{3 \,{\left (2 \, a^{5} b c - a^{6} d\right )} \log \left (b x + a\right )}{b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}} + \frac{3 \,{\left (2 \, b^{2} c^{6} - 6 \, a b c^{5} d + 5 \, a^{2} c^{4} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}} + \frac{7 \, a b^{5} c^{7} - 11 \, a^{2} b^{4} c^{6} d - 2 \, a^{6} c^{2} d^{5} + 2 \,{\left (4 \, b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} - a^{6} d^{7}\right )} x^{2} +{\left (7 \, b^{6} c^{7} - 3 \, a b^{5} c^{6} d - 12 \, a^{2} b^{4} c^{5} d^{2} - 4 \, a^{6} c d^{6}\right )} x}{2 \,{\left (a b^{7} c^{5} d^{5} - 3 \, a^{2} b^{6} c^{4} d^{6} + 3 \, a^{3} b^{5} c^{3} d^{7} - a^{4} b^{4} c^{2} d^{8} +{\left (b^{8} c^{3} d^{7} - 3 \, a b^{7} c^{2} d^{8} + 3 \, a^{2} b^{6} c d^{9} - a^{3} b^{5} d^{10}\right )} x^{3} +{\left (2 \, b^{8} c^{4} d^{6} - 5 \, a b^{7} c^{3} d^{7} + 3 \, a^{2} b^{6} c^{2} d^{8} + a^{3} b^{5} c d^{9} - a^{4} b^{4} d^{10}\right )} x^{2} +{\left (b^{8} c^{5} d^{5} - a b^{7} c^{4} d^{6} - 3 \, a^{2} b^{6} c^{3} d^{7} + 5 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9}\right )} x\right )}} + \frac{b d x^{2} - 2 \,{\left (3 \, b c + 2 \, a d\right )} x}{2 \, b^{3} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.77568, size = 2142, normalized size = 10.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.3167, size = 1182, normalized size = 5.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21052, size = 838, normalized size = 4.23 \begin{align*} -\frac{a^{6} b^{5}}{{\left (b^{12} c^{3} - 3 \, a b^{11} c^{2} d + 3 \, a^{2} b^{10} c d^{2} - a^{3} b^{9} d^{3}\right )}{\left (b x + a\right )}} + \frac{3 \,{\left (2 \, b^{3} c^{6} - 6 \, a b^{2} c^{5} d + 5 \, a^{2} b c^{4} d^{2}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{5} - 4 \, a b^{4} c^{3} d^{6} + 6 \, a^{2} b^{3} c^{2} d^{7} - 4 \, a^{3} b^{2} c d^{8} + a^{4} b d^{9}} - \frac{3 \,{\left (2 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4} d^{5}} + \frac{{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7} - \frac{4 \,{\left (b^{6} c^{5} d^{2} - 2 \, a b^{5} c^{4} d^{3} - 2 \, a^{2} b^{4} c^{3} d^{4} + 8 \, a^{3} b^{3} c^{2} d^{5} - 7 \, a^{4} b^{2} c d^{6} + 2 \, a^{5} b d^{7}\right )}}{{\left (b x + a\right )} b} - \frac{18 \, b^{8} c^{6} d - 54 \, a b^{7} c^{5} d^{2} + 45 \, a^{2} b^{6} c^{4} d^{3} + 20 \, a^{3} b^{5} c^{3} d^{4} - 75 \, a^{4} b^{4} c^{2} d^{5} + 54 \, a^{5} b^{3} c d^{6} - 13 \, a^{6} b^{2} d^{7}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{6 \,{\left (2 \, b^{10} c^{7} - 8 \, a b^{9} c^{6} d + 11 \, a^{2} b^{8} c^{5} d^{2} - 5 \, a^{3} b^{7} c^{4} d^{3} - 5 \, a^{4} b^{6} c^{3} d^{4} + 9 \, a^{5} b^{5} c^{2} d^{5} - 5 \, a^{6} b^{4} c d^{6} + a^{7} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}{\left (b x + a\right )}^{2}}{2 \,{\left (b c - a d\right )}^{4} b^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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